Free Access
Issue
Eur. Phys. J. Appl. Phys.
Volume 88, Number 2, November 2019
Article Number 20904
Number of page(s) 9
Section Physics of Energy Transfer, Conversion and Storage
DOI https://doi.org/10.1051/epjap/2019190284
Published online 05 February 2020

© EDP Sciences, 2020

1 Introduction

The International System of Units (SI) was revised and the revision came into effect from May 20, 2019. This has been decided by the General Conference on Weights and Measures (CGPM) in November 2018 at its 26th meeting [1,2]. From that date, the SI base units are defined via fixed fundamental constants [3,4]. The SI unit of the mass, the kilogram, is now defined via the Planck constant h, and the mole, the SI unit of the amount of substance, is defined via the Avogadro constant NA. The best experimental methods to date to realize and disseminate these fundamental constants are the Kibble balance (watt balance) for the determination of h by comparing electrical power with mechanical power [5] and the X-ray-crystal-density (XRCD) method − using silicon spheres which are chemically ultrapure and highly enriched in 28Si for the determination of the Avogadro constant NA [68]. In Metrology, the most appropriate methods are indicated as “primary methods” yielding the respective quantities with lowest associated uncertainties.

By counting silicon atoms (XRCD method), NA is measured using a sphere of enriched silicon with an approximate mass of 1 kg by(1)(with 8 atoms in the unit cell, the sphere volume V, its mass m, the lattice parameter a, and the molar mass M). N denotes the number of Si atoms in the sphere and n the respective amount of substance [8]. After the revision of the SI, NA is fixed numerically and its associated measurement uncertainty is set to zero by definition. When a new crystal material or sphere in the future will be available, all parameters in the centre of equation (1) have to be determined with lowest associated uncertainties using the XRCD method. The common route used on that level for the dissemination of the SI units kilogram and mol will be as follows: the characterized Si sphere can be used as a primary standard for the mass m (with an uncertainty) according to(2)(with the Rydberg constant R, the speed of light in vacuum c, the fine structure constant α, the Planck constant h, the relative atomic mass of the electron Ar(e), the relative atomic masses Ar(iSi) of the silicon isotopes, the surface layer mass msur, and the mass of crystal point defects mdef). The amount of substance n is disseminated on the highest level via(3)(with the molar mass constant Mu). Moreover, equation (3) shows the relation between h and NA. The production of several highly enriched silicon spheres (from different crystal ingots, each from different production processes) thus serves as a way to generate a pool of primary standards for the mass and the amount of substance used for their dissemination with lowest uncertainties, accessible for science and industry.

In this article, we present the measurement of the molar mass M and isotopic composition x(iSi) of a new highly enriched silicon crystal (Si28-24Pr11), finished after the deadline for the data used for the revision of the SI. Thus, the data are important for the validation, verification and dissemination of m and n. The new crystal shown in Figures 1 and 2 has been produced during a joint-project (“kilogram-2” or “kg-2”) of PTB and Russian institutes and companies [9,10]. We report on the measurement of 17 discrete silicon samples (each with a gross weight of approximately 500 mg) from five different axial positions in the original ingot bracketing the location of the two spheres intended for the XRCD measurements. On each axial position, up to six adjacent radially arranged samples were measured. This could enable an assessment of the variation of M and x(iSi) as a function of the origin in the crystal.

The data evaluation is based on an uncertainty analysis according to the “Guide to the Expression of Uncertainty in Measurement” (GUM) [11]. The results are compared to the two other available enriched crystals Si28-10Pr11 and Si28-23Pr11 which were characterized previously [12,13].

At present, six spheres from three different crystals highly enriched in 28Si have been completely characterized using the XRCD method (two spheres from each crystal). Starting in 2007, the first two enriched spheres (AVO28-S5 and AVO28-S8) from the crystal Si28-10Pr11 were available. In the “kilogram-2” project, two spheres (Si28kg01a and Si28kg01b from the crystal Si28-23Pr11, available in 2015) and another two spheres (Si28kg02a and Si28kg02b from the crystal Si28-24Pr11, available in 2016, this work) have been produced. In the near future, six more enriched spheres (“kilogram-3” project) will be available, finally yielding a set of twelve enriched silicon spheres. Theoretically, the uncertainty associated with the molar mass should be smaller with increasing enrichment in the 28Si isotope as discussed in [13]. However, a higher enrichment might complicate the measurement and thus slightly increase the uncertainty as is shown in this work, meaning from the point of view of the molar mass determination there is an optimum enrichment.

thumbnail Fig. 1

Photograph of the final single crystalline silicon crystal ingot Si28-24Pr11 (length: 52 cm, mass: 5575 g) used for the spheres Si28kg02a and Si28kg02b.

thumbnail Fig. 2

Final silicon crystal ingot (Si28-24Pr11). Left: in the float zone apparatus at the Institute of Crystal Growth (IKZ), Berlin, Germany; right: main parts already cut (the large parts are reserved for the two spheres).

2 Molar mass via isotope ratio measurements

Isotope ratio measurements via high resolution mass spectrometry offer the best access to analyte concentrations (mass fractions) or isotope distributions via amount-of-substance fractions x or even to a molar mass M. In case of the enriched silicon material, all three natural isotopes 28Si, 29Si, and 30Si have to be taken into account, although the two latter are by six and seven orders of magnitude less abundant in the crystal. To overcome the technical problems of absolute measurements of these isotopes in trace amounts and to obtain sufficiently low uncertainties, a special modified method of the classical isotope dilution mass spectrometry (IDMS) technique has been developed [14,15]. This technique denoted as “virtual element” (VE-IDMS) is now routinely applied to determine the molar mass of enriched silicon yielding relative uncertainties urel(M) < 5 × 10−9. In combination with a high-resolution MC-ICP-MS, this method guarantees most precise and accurate results of both the isotopic composition and the molar mass and has been applied and validated by various national metrology institutes (NMIs) [1619]. The measured “intensity” ratios are biased by nature due to space charge effects in the plasma ion source. Therefore, the measured ratios have to be corrected by calibration factors (K factors). Parallel to the VE-IDMS method, an analytical closed form approach has been developed for the absolute determination of the respective K factors which rendered the molar mass measurement of enriched silicon a primary method [12,20]. For a better understanding, the VE-IDMS principle is briefly described.

To avoid the standard measurements of isotope ratios e.g. 30Si/28Si associated with very large uncertainties, in the VE-IDMS method almost only the ratios R = n(30Si)/n(29Si) need to be measured. The silicon material is handled as consisting theoretically of the isotopes 29Si and 30Si only (“impurities”) in the matrix of the most abundant 28Si. The enriched sample material (x) is blended with a silicon material enriched in 30Si (y, spike). The ratio 29Si/30Si has to be measured in the sample, spike, and blend. Additionally, the two masses myx and mx (solid spike material y and sample material x in the blend bx) have to be determined. The molar mass M is calculated via equation (4) (with Rj,2 = xj(30Si)/xj(29Si) and Rj,3 = xj(28Si)/xj(29Si)) (4)

The M(iSi) in equation (4) denotes the respective molar masses derived from the respective atomic masses given in [21]. Using a characterized material (silicon with natural isotopic composition: material w) only the calibration factor K2 had to be calculated. This was performed by the measurement of Rw,2meas during an actual measurement using the known “true” isotope ratio Rw,2true.

3 Experimental

The new crystal material analysed in this work has been manufactured in Russia and finally in Berlin [9,10]. In order to get information about possible variations of the molar mass and isotopic composition, samples (of an almost cubic shape) with an approximate mass of 500 mg each had to be cut from the original ingot (Figs. 3 and 4). The apparent large mass of a single sample is necessary due to the low abundance of the isotopes 29Si and 30Si. The ability of detecting the respective isotope ratios with sufficiently small uncertainties (<2%) requires total silicon mass fractions w(Si) > 4000 μg/g from one sample, which is comparably high when operating ICP-MS instruments. Two samples (L.1.1 and L.1.2) were taken from the tip of the ingot, four samples (N.2.1–N.2.4) were cut right before the origin of first sphere. The samples from part S (S.5.1–S.5.6) bracket the second sphere together with the samples V.1.1.1-V.1.1.5. and V.1.2.1 (which is on top of V.1.1.1). During the measurements the samples V.1.1.3 and V.1.1.4 showed very significant offsets from the other results due to possible contamination with natural silicon or even systematic errors during the sample preparation. Therefore, these two samples were not considered in this investigation. A single sample W.2.1 was cut from the end of the crystal (the so-called Czochralski-region).

The processes of sample preparation and mass spectrometric measurements have been described in detail elsewhere [12,13,22]. For a better understanding, only the main procedures are mentioned. All silicon samples were treated in exactly the same way to enable the possible identification of differences in the isotopic composition and molar mass due to the origin of the sample. All samples were cleaned and the oxide layer was removed by an etching process. After exact weighing, the samples were dissolved in aqueous tetramethylammonium hydroxide (TMAH, mass fraction of the final sample solution after dilution: w(TMAH) = 0.0006 g/g). The concentrations (mass fractions) of the solutions ready for the mass spectrometric measurements were: 4000 μg/g ≤ w(x) ≤ 5000 μg/g, 2500 μg/g ≤ w(bx) ≤ 3500 μg/g, and w(w) = 4 μg/g.

The isotopic composition and molar mass measurements of the silicon crystal samples have been performed using a high resolution multicollector-inductively coupled plasma mass spectrometer (MC-ICP-MS) Neptune™ (Thermo Fisher Scientific GmbH, Bremen, Germany) [22,23]. The operating conditions and parameters are given in Table 1.

Figure 5 shows the plasma of the ICP-MS source via a new bonnet made of sapphire for a better reduction of a possible sample contamination with natural silicon from the ion source. Additional advantages of the sapphire bonnet are the increased mechanical stability compared to a boron nitride bonnet and the fact that it is transparent simplifies the monitoring of the plasma.

The mass spectrometric measurements were performed identical for each sample. Three different “samples” have to be measured in a sequence: four times in the unspiked sample x (enriched silicon) the intensity ratios (in volt) of the isotopes 29Si/30Si are measured. Prior to each sample solution, a blank solution containing aqueous TMAH (w(TMAH) = 0.0006 g/g) was measured in exactly the same way as the sample. These data were subsequently subtracted from the sample data to correct for blank contamination effects. Subsequently, the blend bx followed by the natural silicon solutions w was measured in the same way. K factors for mass bias correction were determined by measuring the isotope ratios Rw,2 = Iw(30Si)/Iw(29Si) in the “calibration” solution (natural silicon w, material name: WASO04) at the end of the IDMS-sequence to avoid a cross contamination of the enriched solutions with natural silicon.

The K factor was determined by the ratio of the correct (“true”) isotope ratios determined in [12] and the measured isotope ratios Rw,2 = Iw(30Si)/Iw(29Si) in the solution w of each sequence.

thumbnail Fig. 3

Schematic cross section of the silicon crystal Si28-24Pr11. The samples measured were taken from parts L, N, S, V, and W. The regions N, S, and V are bracketing the two spheres in the segments P and T. Each sample (cubic shape) has an approximate mass of 500 mg.

thumbnail Fig. 4

Cross section of the disc of part V. In the lower centre region, five adjacent samples V1.1.1–V1.1.5 are indicated (compare Fig. 3).

Table 1

Operating conditions of the MC-ICP-MS with components almost silicon-free (PFA: perfluoroalkoxy alkane, PEEK: polyether ether ketone).

thumbnail Fig. 5

Inductively-coupled plasma visible via a new sapphire bonnet type with reduced content of silicon with natural isotopic composition.

4 Results and discussion

The samples schematically indicated in Figure 3 were measured under as equal as possible conditions. The aim was to determine the respective molar masses M and the isotopic compositions expressed in their amount-of-substance fractions x( i Si) (i = 28, 29, 30) throughout the original silicon crystal ingot Si28-24Pr11 in a most comparable way. Similar studies have been conducted on the crystals Si28-10Pr11 (“AVO28”) [12] and Si28-23Pr11 [13]. The present crystal however, has the highest enrichment in 28Si and therefore, at a first glance, a respectively lowered uncertainty associated with M was expected. But due to the increased difficulties with the measurement of the extremely low 30Si signal an increased uncertainty was also possible. Each sample is usually measured six times (six sequences). In contrast to previous studies, more axial positions were chosen to have a more significant local difference of the sample origin. 17 samples have been measured at PTB, resulting in an average molar mass M = 27.976 933 787(77) g/mol with a relative combined uncertainty uc,rel(M) = 2.7 × 10−9.

This result already covers the scattering of the averaged results of the individual samples as shown in Figure 6.

As can be clearly seen, an additional uncertainty contribution from the scattering of the values must be added in contrast to previous studies (included in the relative uc,rel(M) = 2.7 × 10−9 (k =1; indicated by dashed lines in Fig. 6). The relative uncertainty without the scattering contribution is only uc,rel(M) = 1.2 × 10−9 (k =1; indicated by dotted black lines in Fig. 6).

The scattering effect has a contribution of 56% to the overall uncertainty associated with M.

Note that this plot shows only a two-dimensional relation. Therefore, the uncertainty of the molar mass of the crystal Si28-24Pr11 is ranging between the two previously measured crystals: urel(M, Si28-10Pr11) =4.4 × 10−9, and urel(M, Si28-23Pr11) = 1.4 × 10−9 [12,13]. Usually, it is theoretically expected that the higher the enrichment x(28Si), the smaller the respective uncertainty associated with M. For the current crystal Si28-24Pr11 x(28Si) = 0.999 993 104(66) mol/mol has been determined. This is the highest enrichment of a silicon crystal intended for the use by the XRCD method, so far. The individual combined uncertainties are to some extent <10−9. However, the significant scattering of the results of the different samples had to be taken into account.

Figure 7 displays three-dimensional molar mass plots for a more realistic picture of the origin of the samples (given by their axial and radial positions in mm) and their respective molar masses. In parts a and b of Figure 7, the average molar mass values of the individual samples (with uncertainties) are plotted (which is a more descriptive but less precise view than the plot in Fig. 6). However, the detailed sample indication is omitted for more clarity (b is a tilted version of a). Part c of Figure 7 is a top view on the axis origins (radial and axial) of the different samples. Axial and radial positions of the samples are given in Table 2 (part W: not shown).

To check the data for the presence of any kind of inhomogeneity, the concept of degrees of equivalence di (a common consistency check) was applied. For the successful application by the XRCD method it can be shown that the individual molar masses are consistent with the respective average molar mass within their limits of uncertainty. Equation (5) defines the di as the difference of the N =17 individual values Mi and the respective average M:(5)

The corresponding uncertainty is given by(6)

The di were calculated using their individual uncertainties only (not including an uncertainty contribution due to the scattering). When an individual di is smaller than its respective uncertainty, the corresponding value is consistent with the overall average.

This can be visualized by the di , and their associated expanded uncertainties U(di ) with k = 2, shown in Figure 8.

In Figure 8, five samples (L1.1; S.5.3; V.1.1.2, V.1.1.5, W.2.1 very slightly) do not cover the zero line as a criterion for consistency. This is subsequently taken into account by adding the type A uncertainty of M calculated from the 17 individual results.

The combined uncertainty of M was then calculated using equation (7) (7)with the standard deviation s based on [2426]. Although the scattering of the values in Figure 6 is significant, the DoE analysis shown in Figure 8 clearly suggests a consistent and homogeneous property of all samples. Because of the comparably extreme enrichment in 28Si, the slightly different measurement conditions show an impact on the results indicated by the scattering. As an example, the six measurements (sequences) of sample L.1.1 are displayed in Figure 9.

From Figure 9 it can be clearly deduced that scattering between different measurements appears even for an individual sample. This is not due to inhomogeneities, but it is caused by different (although tiny) experimental variations which cannot be specified. The same holds true for the comparison of the different samples which scatter in the same order of magnitude as in a single sample shown in Figure 9. For this reason, a material dependent inhomogeneity cannot be confirmed. In contrast, a homogeneous behaviour of the molar mass according to the sample origin is more plausible when considering the DoE analysis shown in Figure 8 (where the individual uncertainties without the additional contribution of the scattering were used).

As an example, a representative uncertainty budget of a single measurement of the molar mass M of sample S.5.6 using equation (4) as a model equation is given in Table 3. The uncertainty calculation using the GUM Workbench Pro™ software (version 2.4.1 392; Metrodata GmbH, Germany) has been done according to the GUM [11].

The largest uncertainty contribution to the uncertainty of M is , the measured intensity ratio 29Si/30Si in the blend bx with a contribution of 53%. The next notable contribution originates from Rw,2, the corrected isotope ratio 29Si/30Si in the natural “calibration” solution w. The measured intensity ratio (29Si/30Si) in the sample contributes only another 5%. The combined relative uncertainty (k = 1) of this very sample has been determined to urel(M) = 8.7 × 10−10.

Table 4 summarizes the molar masses M and amount-of-substance fractions x(iSi) obtained from measurement campaigns during the last years.

In the current study, the Si crystal has the highest enrichment in 28Si with x(28Si) > 0.999 99 mol/mol. As predicted in a previous study [13], the higher x(28Si) the smaller the associated uncertainty and the respective uncertainty of M. However, the average uncertainty associated with M in this study (Si28-24Pr11) increases, compared to the crystal measured in the previous study (Si28-23Pr11) due to additional uncertainty contributions from data scattering. Since the average uncertainties associated with the molar mass are <5 × 10−9, all six spheres produced from the three crystals are suitable for the use in the XRCD method and moreover suitable as primary reference standards for the dissemination of the kilogram and the mole in the future.

thumbnail Fig. 6

Average values of molar masses of the 17 individual crystal samples of the new crystal Si28-24Pr11 measured in the current study. Combined uncertainties (k = 1) are indicated by error bars. The arithmetic mean is M = 27.976 933 787(77) g/mol (indicated by the solid line). The relative combined uncertainty associated with the average molar mass is uc,rel(M) = 2.7 × 10−9 (including the contribution of scattering). Upper and lower limits of this uncertainty are shown by red dashed lines. The dotted black lines indicate the upper and lower uncertainty limits without the contribution of scattering.

thumbnail Fig. 7

Three-dimensional molar mass plots showing the axial and radial origins of the samples together with the respective molar masses. The plots serve as a rough indication of the sample origin in the initial crystal ingot (Fig. 1). Detailed molar mass distributions are shown in Figure 6.

Table 2

Exact positions of the silicon samples in the crystal and respective molar masses.

thumbnail Fig. 8

Degrees of equivalence di of the results of the average values of the molar mass of the 17 crystal samples. Error bars denote the expanded uncertainties (k = 2) associated with the di . Note that data encompassing the zero line with their uncertainties are consistent with the average molar mass. This uncertainty still does not cover the scattering described in the text.

thumbnail Fig. 9

Individual measurements (sequences) of the molar mass of the crystal sample Si28-24Pr11L.1.1 with the individual combined uncertainty (k =1). The solid line represents the average molar mass of sample L.1.1 with upper and lower limits of the average uncertainty (dashed lines).

Table 3

Sample S.5.6: representative uncertainty budget of a single measurement of M.

Table 4

Arithmetic mean values of M, x(28Si), x(29Si), and x(30Si) of the different available Si crystals highly enriched in 28Si. Numbers in brackets denote uncertainties in the last digits (k = 1).

5 Conclusion

We report on the examination of the distribution of the molar mass M and the isotopic composition expressed by the respective amount-of-substance fractions x(iSi) measured in a new silicon crystal highly enriched in 28Si. This is the third crystal ever available to disseminate the mole and the kilogram via the XRCD method providing two additional silicon spheres usable as primary reference standards for this purpose. The new crystal with the code Si28-24Pr11 shows no significant distribution of the molar mass with respect to the origin of the different measured samples within the limits of uncertainty. The bias of single values can be clearly traced back to experimental scattering. The latter is more evident in this crystal material, because of the extreme enrichment in 28Si (x(28Si) > 0.999 99 mol/mol) which is associated with a respective reduced uncertainty (without the statistical contribution). The average relative combined uncertainty of the molar mass uc,rel(M) = 2.7 × 10−9 legitimates the use of this new crystal material being implemented in the available very limited pool of enriched silicon spheres as primary reference standards.

Acknowledgments

The authors gratefully acknowledge discussions with Stefan Wundrack (PTB) and Nikolay Abrosimov (IKZ). Many thanks to Daniela Eppers (PTB) for providing the samples from distinct crystal locations.

Author contribution statement

Both authors contributed equally to the experiments and writing of this paper.

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Cite this article as: Axel Pramann, Olaf Rienitz, The molar mass of a new enriched silicon crystal: maintaining the realization and dissemination of the kilogram and mole in the new SI, Eur. Phys. J. Appl. Phys. 88, 20904 (2019)

All Tables

Table 1

Operating conditions of the MC-ICP-MS with components almost silicon-free (PFA: perfluoroalkoxy alkane, PEEK: polyether ether ketone).

Table 2

Exact positions of the silicon samples in the crystal and respective molar masses.

Table 3

Sample S.5.6: representative uncertainty budget of a single measurement of M.

Table 4

Arithmetic mean values of M, x(28Si), x(29Si), and x(30Si) of the different available Si crystals highly enriched in 28Si. Numbers in brackets denote uncertainties in the last digits (k = 1).

All Figures

thumbnail Fig. 1

Photograph of the final single crystalline silicon crystal ingot Si28-24Pr11 (length: 52 cm, mass: 5575 g) used for the spheres Si28kg02a and Si28kg02b.

In the text
thumbnail Fig. 2

Final silicon crystal ingot (Si28-24Pr11). Left: in the float zone apparatus at the Institute of Crystal Growth (IKZ), Berlin, Germany; right: main parts already cut (the large parts are reserved for the two spheres).

In the text
thumbnail Fig. 3

Schematic cross section of the silicon crystal Si28-24Pr11. The samples measured were taken from parts L, N, S, V, and W. The regions N, S, and V are bracketing the two spheres in the segments P and T. Each sample (cubic shape) has an approximate mass of 500 mg.

In the text
thumbnail Fig. 4

Cross section of the disc of part V. In the lower centre region, five adjacent samples V1.1.1–V1.1.5 are indicated (compare Fig. 3).

In the text
thumbnail Fig. 5

Inductively-coupled plasma visible via a new sapphire bonnet type with reduced content of silicon with natural isotopic composition.

In the text
thumbnail Fig. 6

Average values of molar masses of the 17 individual crystal samples of the new crystal Si28-24Pr11 measured in the current study. Combined uncertainties (k = 1) are indicated by error bars. The arithmetic mean is M = 27.976 933 787(77) g/mol (indicated by the solid line). The relative combined uncertainty associated with the average molar mass is uc,rel(M) = 2.7 × 10−9 (including the contribution of scattering). Upper and lower limits of this uncertainty are shown by red dashed lines. The dotted black lines indicate the upper and lower uncertainty limits without the contribution of scattering.

In the text
thumbnail Fig. 7

Three-dimensional molar mass plots showing the axial and radial origins of the samples together with the respective molar masses. The plots serve as a rough indication of the sample origin in the initial crystal ingot (Fig. 1). Detailed molar mass distributions are shown in Figure 6.

In the text
thumbnail Fig. 8

Degrees of equivalence di of the results of the average values of the molar mass of the 17 crystal samples. Error bars denote the expanded uncertainties (k = 2) associated with the di . Note that data encompassing the zero line with their uncertainties are consistent with the average molar mass. This uncertainty still does not cover the scattering described in the text.

In the text
thumbnail Fig. 9

Individual measurements (sequences) of the molar mass of the crystal sample Si28-24Pr11L.1.1 with the individual combined uncertainty (k =1). The solid line represents the average molar mass of sample L.1.1 with upper and lower limits of the average uncertainty (dashed lines).

In the text

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